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MEN  OE ilATHE^IATICS
a heuristic guide to propositions \vhich may be true, but which
are not necessarily so, even when they have been deduced by a
rigid application of Aristotelian logic, and he says that
numerous false theories (including Cantor's) have been erected
on this rotten foundation during the past half century.
Such a revolution in the rudiments of mathematical thinking
does not go unchallenged. Brouwer's radical move to the left is
speeded by an outraged roar from the reactionary right, 'What
Weyl and Brouwer are doing [Brouwer is the leader, Weyl his
companion in revolt] is mainly following in the steps of Kro-
necker*, according to Hilbert, the champion of the status quo,
4They are trying to establish mathematics by jettisoning every-
thing which does not suit them and setting up an embargo. The
eSect is to dismember our science and to run the risk of losing
part of our most valuable possessions. Weyl and Brouwer
condemn the general notions of irrational numbers, of functions
- even of such functions as occur in the theory of numbers -
Cantor's transfinite numbers, etc., the theorem that an infinite
set of positive integers has a least, and even the "law of ex-
cluded middle", as for example the assertion: Either there is
only a finite number of primes or there are infinitely many.
These are examples of [to them] forbidden theorems and modes
of reasoning. I believe that impotent as Kronecker was to
abolish irrational numbers (Weyl and Brouwer do permit us to
retain a torso), no less impotent will their efforts prove to-day.
No! Brouwer's programme is not a revolution, but merely the
repetition of a futile coup de main with old methods, but which
was then undertaken with greater verve, yet failed utterly.
To-day the State [mathematics] is thoroughly armed and
strengthened through the labours of Erege, Dedekind, and
Cantor. The efforts of Brouwer and Weyl are foredoomed to
futility.'
To which the other side replies by a shrug of the shoulders
and goes ahead with its great and fundamentally new task of re-
establishing mathematics (particularly the foundations of
analysis) on a firmer basis than any laid down by the men of
the past 2,500 years from Pythagoras to Weierstrass,
What will mathematics be like a generation hence when - w&
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